The 4-joint RRRP robot, like you see here, is a popular choice for certain kinds of assembly tasks. In this picture, we see it at its zero configuration. To solve the forward kinematics, we need to find M, the configuration of the {b}-frame, and the joint screw axes when the arm is at its zero configuration. Then we can use the product of exponentials formulas from the previous videos.

First let's focus on the orientation of M. From the picture, we can see that the {b}-frame x-axis is aligned with the minus y-axis of the {s}-frame. The {b}-frame y-axis is aligned with the minus x-axis of the {s}-frame. And the {b}-frame z-axis is aligned with the minus z-axis of the {s}-frame. Also, we can see that the {b}-frame is offset from the {s}-frame by 19 units in the x-direction and -3 units in the z direction of the {s}-frame. We add the row of zeros and a one to complete the M matrix.

Next let's find the screw axis of joint 1, expressed in the {s}-frame. The axis of rotation is aligned with the {s}-frame z-axis, so the angular component of S1 is zero, zero, one. A rotation about this axis causes no linear motion of a point at the origin of the {s}-frame, so the linear component of the screw S1 is zero.

We can also express the screw axis as B_1 in the {b}-frame. The joint axis is in the negative z_b direction, so the angular component is zero, zero, minus one. A unit angular velocity about the joint 1 axis induces a linear velocity at a point at the origin of the {b}-frame, and it is apparent from the figure that this linear velocity is 19 units in the minus x_b direction. This should be readily apparent from the figure; you shouldn't have to do any math.

Now let's go faster through the rest of the joints. Joint 2's axis is aligned with the {s}-frame z-axis, so the angular component of the {s}-frame screw S_2 is zero, zero, one. Unit angular velocity about this axis induces a linear velocity at the {s}-frame origin of 10 units in the minus y_s direction. The screw axis B_2 in the {b}-frame is zero, zero, minus one, minus 9, zero, zero.

Joint 3's rotational screw axis induces a large linear velocity at the origin of the {s}-frame but zero linear velocity at the origin of the {B}-frame.

Finally, the prismatic axis of joint 4 is aligned with the z-axis of the {s}-frame and the minus z-axis of the {b}-frame. The angular component of both screw axes is zero, since it is a prismatic joint.

For most open-chain robots, deriving the screw axes is just this easy: you can simply look at a good drawing of the robot at its zero configuration and get the screw axes by inspection. If anything is unclear about what we did, you should either pause this video at appropriate places or look at the examples in the book.

So, this concludes Chapter 4. You now know how to use the material of Chapter 3 to solve the forward kinematics of robots. In Chapter 5, we will study the velocity kinematics relating joint velocities to the twist of the end-effector.